The class-action complaint, filed in US District Court in Riverside, California, east of Los Angeles, also alleges the automaker tried to cover up hundreds of related accidents that caused injuries and property damage and has dragged its feet in furnishing a remedy for the issue.

The plaintiffs are seeking restitution for the diminished resale value of their Grand Cherokees stemming from the "raft of negative publicity" surrounding the recent disclosure of the rollaway problem.

"In light of the stigma attached to those vehicles by (Fiat Chrysler's) conduct, they are now worth significantly less than they otherwise would be," the lawsuit states.

Company spokesman Michael Palese said the automaker has not yet been served with the lawsuit and it would be inappropriate to comment before reviewing the case.

Fiat Chrysler Automobiles NV issued a recall in April for 1.1 million cars and SUVs worldwide, most of them in the United States, because the vehicles could roll away after drivers exit them.

The recall applies to 811,000 vehicles in the United States, covering 2014-2015 model years of the Jeep Grand Cherokee sport-utility vehicle, and 2012-2014 Dodge Charger and Chrysler 300 sedans.

The lawsuit came as the company announced that software upgrades for all 1.1 million recalled vehicles would be available by the end of this month.

While not acknowledging a design flaw in its transmission gear selectors, the company has said roll aways have occurred because drivers mistakenly believe they had placed the vehicles in park before getting out.

Chrysler Fiat said the issue was linked to 700 incidents overall, including 41 injuries, 212 crashes and 308 reports of property damage.

A recalled 2015 Grand Cherokee was involved in Sunday's death of Yelchin, 27, who was crushed when his vehicle rolled backward on the steep driveway of his Los Angeles home, pinning him against a brick wall and a fence.

Los Angeles police have said their detectives were examining whether the recall issue played a role. If so, the Russian-born actor would be the first fatality reported linked to the recall.

The US National Highway Traffic Safety Administration (NHTSA) has said the gear selector in question clearly poses a safety issue that has led to hundreds of crashes and dozens of injuries. The agency said testing of the electronic shifter found it "not intuitive and provides poor tactile and visual feedback to the driver, increasing the potential for unintended gear selection".